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Search: id:A100936
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| A100936 |
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Symmetric square array, read by antidiagonals, where the inverse binomial transform of row n equals: [C(n,0)*1, C(n,1)*2,..., C(n,k)*A051163(k), ..., C(n,n)*A051163(n)] and where A051162 equals the antidiagonal sums. |
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4
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| 1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 14, 7, 1, 1, 9, 28, 28, 9, 1, 1, 11, 47, 76, 47, 11, 1, 1, 13, 71, 163, 163, 71, 13, 1, 1, 15, 100, 301, 435, 301, 100, 15, 1, 1, 17, 134, 502, 971, 971, 502, 134, 17, 1, 1, 19, 173, 778, 1909, 2577, 1909, 778, 173, 19, 1, 1, 21, 217, 1141
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Antidiagonal sums form A051163. Main diagonal is A100937. Different from A086620.
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FORMULA
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T(n, k) = Sum_{j=0..n} C(k, j)*C(n, j)*A051162(j), with T(0, 0) = 1 and where Sum_{i=0..n} T(n-i, i) = A051162(n).
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EXAMPLE
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Rows begin:
[1,1,1,1,1,1,1,1,1,...],
[1,3,5,7,9,11,13,15,17,...],
[1,5,14,28,47,71,100,134,...],
[1,7,28,76,163,301,502,778,...],
[1,9,47,163,435,971,1909,3417,...],
[1,11,71,301,971,2577,5917,12167,...],
[1,13,100,502,1909,5917,15678,36744,...],
[1,15,134,778,3417,12167,36744,97272,...],...
Antidiagonal sums form A051163: [1,2,5,12,30,76,194,496,1269,3250,8337,...].
The inverse binomial transform of the rows form the respective rows of the triangle B:
[1*1],
[1*1,1*2],
[1*1,2*2,1*5],
[1*1,3*2,3*5,1*12],
[1*1,4*2,6*5,4*12,1*30],...
where B(n,k) = binomial(n,k)*A051163(k).
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PROGRAM
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(PARI) {T(n, k)=if(n==0|k==0, 1, sum(j=0, n, binomial(k, j)*binomial(n, j)*sum(i=0, j, T(j-i, i))); )}
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CROSSREFS
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Cf. A051163, A100937.
Sequence in context: A144461 A106597 A108359 this_sequence A086620 A137897 A056152
Adjacent sequences: A100933 A100934 A100935 this_sequence A100937 A100938 A100939
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Nov 23 2004
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